A golem with movement + rally, accidentally gets into combat.

Does the rally activate at the beginning, and then the golem keeps the panic token for getting into combat, or at the end, so he ends clean?

Bonus actions from modifier dials happen after the first action is completed. So the Rune Golem would move, gain a panic, then rally. (Rule Reference 14.1 is where it says it happens after the main action.)

Yes, it looks clear now. Thank you. Easy way of moving with the golem then.

Note though that this denies the golem the Inspiration token it might be eligible for otherwise.